# Rodion Kuzmin

Template:Infobox scientist Rodion Osievich Kuzmin (Russian: Родион Осиевич Кузьмин{{#invoke:Category handler|main}}, Nov. 9, 1891, Riabye village in the Haradok district – March 23, 1949, Leningrad) was a Russian mathematician, known for his works in number theory and analysis. His name is sometimes transliterated as Kusmin.

## Selected results

$x={\frac {1}{k_{1}+{\frac {1}{k_{2}+\cdots }}}}$ is its continued fraction expansion, find a bound for
$\Delta _{n}(s)=\mathbb {P} \left\{x_{n}\leq s\right\}-\log _{2}(1+s),$ where
$x_{n}={\frac {1}{k_{n+1}+{\frac {1}{k_{n+2}+\cdots }}}}.$ Gauss showed that Δn tends to zero as n goes to infinity, however, he was unable to give an explicit bound. Kuzmin showed that
$|\Delta _{n}(s)|\leq Ce^{-\alpha {\sqrt {n}}}~,$ where C,α > 0 are numerical constants. In 1929, the bound was improved to C 0.7n by Paul Lévy.
$2^{\sqrt {2}}=2.6651441426902251886502972498731\ldots$ is transcendental. See Gelfond–Schneider theorem for later developments.
$\sum _{n\in I}e^{2\pi if(n)}\ll \lambda ^{-1}.$ 